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							Elasticity of solids<span>.</span>
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					<h2><span>大树为什么屹立不倒？——固体的弹性</span></h2>
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									<span class="posted-on">Nov. 29th 2021</span>
									<h3><a href="#">势能的求解</a></h3>
									<span class="category">Elasticity of solids</span></div>
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										总述
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										&emsp;&emsp;
										下面我们来求解势能。一棵弯曲的树有两种势能：重力势能和它自己的弹性势能。在这里我们假设树弯曲的角度很小。我们先从简单的入手，先求重力势能。
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											重力势能的求解
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										&emsp;&emsp;
										由于树弯曲的角度很小，所以可以忽略树的曲率，认为其弯曲时还成直线，如图
										<a href="#"><img class="img-responsive center-block" src="images/树的重心.jpg" alt="" width="300" title="树的重心"></a>
										设弯曲角（图中的小角）为ψ，树的密度是ρ，高度是h，半径是r，则其重力势能变化量是：<br>
										     <a href="#"><img class="img-responsive center-block" src="images/重力势能的计算.jpg" alt="" width="300" title="重力势能的计算"></a>
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										弹性势能的求解
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										&emsp;&emsp;
										可以把树看成圆柱形的弹性体，发生弯曲时，它的弹性势能增加。
										引入杨氏模量E：
										<a href="#"><img class="img-responsive center-block" src="images/弹性势能的计算1.jpg" alt="" width="100" title="弹性势能的计算"></a>
										
										<a href="#"><img class="img-responsive center-block" src="images/弹性势能的计算2.jpg" alt="" width="300" title="弹性势能的计算"></a>
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											如图在顶端圆面上如图取角微元：
											<a href="#"><img class="img-responsive center-block" src="images/弹性势能的计算3.jpg" alt="" width="320" title="弹性势能的计算"></a>
											则元底面积：
											<a href="#"><img class="img-responsive center-block" src="images/弹性势能的计算4.jpg" alt="" width="200" title="弹性势能的计算"></a>
											以元底面积为底的近似长方体微元被压缩的长度：
											<a href="#"><img class="img-responsive center-block" src="images/弹性势能的计算5.jpg" alt="" width="150" title="弹性势能的计算"></a>
											而根据弹性体的弹性势能计算公式：
											<a href="#"><img class="img-responsive center-block" src="images/弹性势能的计算6.jpg" alt="" width="150" title="弹性势能的计算"></a>
											整理以上算式可以得到弹性势能：
											<a href="#"><img class="img-responsive center-block" src="images/弹性势能的计算7.jpg" alt="" width="300" title="弹性势能的计算"></a>
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									总结
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									把上述两项势能相加得总势能变化量关于弯曲角度的函数（运用了小角近似）：
									<a href="#"><img class="img-responsive center-block" src="images/总结.jpg" alt="" width="350" title="总结"></a>
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